The take-home message of Theorem REF  is that using \(=\)  is preferable, because it retains the computational advantages of \(\)  and the key mathematical properties of \(\)  w.r.t. the weak* topology, which are both desirable to train models in practice.
Indeed, as discussed in [1]}, [2]}, the additional terms \((,)\)  and \((,)\)  require even less computation time than \((,)\) .
Thus computing \((,)\)  takes almost as much time as \(_\epsilon (,)\) .
Furthermore, the properties of the Sinkhorn divergence hold for any \(\epsilon >0\) , which allows to consider \(\)  as an interesting object in itself, not only as a proxy for unregularized OT when \(\epsilon \)  is small.
